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I don't know if this integral was in the literature, then refers it please. This question is a curiosity after I did some experiments with Wolfram Alpha online calculator, and I hope that has mathematical meaning, that is, that I believe that it is well-defined.

Consider the sequence of integrals with first term
$$\int_1^{\infty}x^{-x}dx,$$ and second $$\int_1^{\infty}\int_1^{\infty}x^{-y}y^{-x}dxdy.$$

And now the third term of this sequence being a triple integral, with factors in this new integrand $x^{-y}, y^{-x}$ and $z^{-x}, z^{-y}$, and also $z^{-x}$ and $z^{-y}$.

Thus I am saying this integral $$\int_1^{\infty}\int_1^{\infty}\int_1^{\infty}x^{-y-z}y^{-x-z}z^{-x-y}dxdydz.$$ And the next integral of our infinite sequence of integrals, with a similar pattern than previous.

Question. Is it known what is the value of the limit integral? If you know the literature please refers it. If it isn't in the literature, can you then calculate an approximation of such limit integral? Or well a good upper bound. Many thanks.

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    $\begingroup$ Oh dear, save me from this madness... $\endgroup$
    – Simply Beautiful Art
    Jul 18, 2017 at 19:13
  • $\begingroup$ The first integral is discussed here math.stackexchange.com/questions/2358024/… ... it is not Sophomore's dream ! $\endgroup$
    – Donald Splutterwit
    Jul 18, 2017 at 19:14
  • $\begingroup$ Wait, you are asking for the limit of these integrals? $\endgroup$
    – Simply Beautiful Art
    Jul 18, 2017 at 19:14
  • $\begingroup$ Yes I know @DonaldSplutterwit many thanks. $\endgroup$
    – user243301
    Jul 18, 2017 at 19:18
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    $\begingroup$ It is a good exercice for you. So instead of waiting other people work for you, what did you try ? $\endgroup$
    – reuns
    Jul 18, 2017 at 19:59

1 Answer 1

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Conjecture: they tend to zero:

$\int_1^{\infty } x^{-x} \, dx\approx 0.70417$

$\int _1^{\infty }\int _1^{\infty }x^{-y} y^{-x}dydx\approx 1.2259$

$\int _1^{\infty }\int _1^{\infty }\int _1^{\infty }x^{-y-z} y^{-x-z} z^{-x-y}dzdydx\approx 0.120842$

$\int _1^{\infty }\int _1^{\infty }\int _1^{\infty }\int _1^{\infty }x^{-w-y-z} y^{-w-x-z} z^{-w-x-y} w^{-x-y-z}dwdzdydx\approx 0.0116407$

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  • $\begingroup$ Many thanks for this new contribution. $\endgroup$
    – user243301
    Jul 18, 2017 at 22:44

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